I love the sort of thin omelettes the Italians sometimes turn into sandwiches: cold and pressed between two pieces of mayonnaised bun or slices of schiacciata. And that's what I'd warmly advise here in the unlikely event that you end up with leftovers. But they are so good hot and straight off the press, eaten either with knife and fork or rolled up within a warmed tortilla, that I urge you to morph into a kind of short order omelette cook next time you have a batch of people to feed convivially in the morning or not long after. The way to make this easy is, first get a good pan (I like a Scanpan Crepe Pan) and then get out loads of eggs and leave them out near a mixing bowl by the cooker, and then mix up a few ideas for fillings and set them out in their bowls nearby. Then, all you do is crack 2 eggs, add your filling, fry, toss out and get on with the next. I've jotted down what I put in the omelettes. Obviously, I don't expect you to be restricted, but I thought it might be helpful.
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Heat a crepe pan or heavy based frying pan/skillet with the butter and oil. Once the pan is hot, pour in the egg mixture swirling quickly to get an even and thin coating in the pan. Let the omelette cook for a couple of minutes over medium-high heat.
Lift the edge of the omelette with a spatula to check it is set and golden underneath; the top of the omelette should be just-about-set but still a little gooey.
Slip the omelette out of the pan onto a plate and flip 1/2 of the omelette over the other, or fold in 3 like a business letter. Carry on!